I believe that since the two rolls are independent from one another, the covariance/correlation should be 0. Increasing the number of dice sides would not change this relationship, thus the covariance should stay the same. This is all theoretical, of course, so experiments may come to different conclusions.
Edit: Please check out this comment. Apparently the question should be phrased as How does increasing the number of sides impact the absolute value of the covariance? instead.